Non Standard Logics & Modal Logics

Models and Formalisms
S. Garlatti
MR2A Informatique
2012

Progress
1 Introduction
2 Modal Logics
3 Model Theory
4 Axiomatic Theory
5 Main Modal Systems
6 Axioms and Class of Models
7 A Knowledge and Belief Logic
2 / 41 SG Models and Formalisms

Non-classical Logics
Classical logics: propositional logic, first-order logic
Non-classical logics can be classified in two main categories:
Extended logics
New logical constants are added
The set of well-formed formulas (wff) is a proper superset of the
set of well-formed formulas in classical logic.
The set of theorems generated is a proper superset of the set of
theorems generated by classical logic,
Added theorems are only the result of the new wff.

Classical logics: propositional logic, first-order logic
Non-classical logics can be classified in two main categories:
Deviant logics
The usual logical constants are kept, but with a different meaning
Only a subset of the theorems from the classical logic hold
A non-exclusive classification, a logic could be in both.

Extended Logics
Modal Logics
Tense Logics
Combination of Tense and Modality
Intensional Logic
Dynamic Logic
Deontic Logic (obligatory, permitted)
Conditional logic
...

Deviant Logics
Intuitionistic logics
Multivalued logics
Fuzzy logics
Paraconsistent logics
Probabilistic logic
...

Progress
1 Introduction
2 Modal Logics
3 Model Theory
4 Axiomatic Theory

Modal Logics
Traditional alethic modalities or modalities of truth
Propositional logic
New logical constants
Necessity:
Possibility: ♦
Alphabet
A set of propositional symbols S = {p1, p2, ..., pn}
A set of connectors: ¬, →, ↔, ∨, ∧
A set of modalities: , ♦

The Language
Well-Formed Formulae
Let pi be a propositional symbol, (pi ∈ S), pi is an Atomic
formula
If G and H are wff
(G → H), (G ↔ H), (G ∨ H), (G ∧ H), ¬G, G and ♦G
are wff .
Let P and Q be wff
( P → ♦P)
( ♦P → ♦♦Q)
(( ♦P ∨ ♦ ♦Q) ∧ P)

Progress
1 Introduction
2 Modal Logics
3 Model Theory
4 Axiomatic Theory

Model Theory
In Propositional and Predicate Logics
The Interpretation function I is truth-functional: the truth or
falsity of a well-formed formula is determined by the truth or
falsity of its components.

Model Theory
In Modal Logics, is-it the same?
P ¬P P
V F ?
F V F

Model Theory
In Modal Logics, is-it the same?
P ¬P P
V F ?
F V F
In Modal logics, the Interpretation function I is not
truth-functional

Model Theory
A standard Model is a structure M = (W, R, P) where
W is a set of possible worlds

Model Theory
R is a binary relation on W (R ⊆ W × W)

Model Theory
R is an accessibility relationship between possible worlds.
We can write: α R β

Model Theory
P is a mapping from natural numbers to subsets of W
(Pn ⊆ W, for each natural number n)

Model Theory
P is a mapping from natural numbers to subsets of W
(Pn ⊆ W, for each natural number n)
P is an assignement of truth value to atomic formulae at possible
worlds.
It is a function on the set {0, 1, 2, ..., } of natural numbers such
that for each such number n, Pn is a subset of W.
It represents an assignment of sets of possible worlds to atomic
formulae (for pi ∈ S, Pi ⊆ W, ∀α such that α ∈ Pi , pi is true),

Possible Worlds, Truth, Validity
Truth in a possible world, α ∈ W in M = (W, R, P)

For propositional symbols pi , |=M
α pi iﬀ α ∈ Pi for
n = {1, 2, 3, ..., n}

n = {1, 2, 3, ..., n}
|=M
α A means that A is true for the world α in the standard
model M

n = {1, 2, 3, ..., n}
|=M
model M
|=M
α ¬A iﬀ not |=M
α A

n = {1, 2, 3, ..., n}
|=M
model M
|=M
α ¬A iﬀ not |=M
α A
|=M
α (A ∨ B) iﬀ either |=M
α A or |=M
α B), or both

n = {1, 2, 3, ..., n}
|=M
model M
|=M
α ¬A iff not |=M
α A
|=M
α (A ∨ B) iff either |=M
α A or |=M
α B), or both
|=M
α (A ∧ B) iff both |=M
α A and |=M
α B)

|=M
α A iﬀ for every β in M such that αRβ, |=M
β A

|=M
β A
|=M
α ♦A iﬀ for some β in M such that αRβ, |=M
β A

|=M
β A
|=M
β A
Truth in a Model
|=M
A iﬀ for every world α in M, |=M
α A

Progress
1 Introduction
2 Modal Logics
3 Model Theory
4 Axiomatic Theory

Axiomatic Theory
Modal Logic KT
Axioms

Axiomatic Theory
Modal Logic KT
Axioms
Necessity axiom, called T
( P → P)

Axiomatic Theory
Modal Logic KT
Axioms
( P → P)
Axiom K
( (P → Q) → ( P → Q))

Axiomatic Theory
Modal Logic KT
Axioms
( P → P)
Axiom K
( (P → Q) → ( P → Q))
Inference Rules
Necessity Rule
P P

Axiomatic Theory
Modal Logic KT
Axioms
( P → P)
Axiom K
( (P → Q) → ( P → Q))
Inference Rules
Necessity Rule
P P
Modus Ponens
P, (P → Q) Q

KT System: Axiomatic Theory
Properties of the Modal Logic KT
Consistency: KT is Consistent
Soundness: KT is Sound
Completeness: KT is Complete
Consequences: Axioms K and T are Valid Formulae.

Progress
1 Introduction
2 Modal Logics
3 Model Theory
4 Axiomatic Theory

Main Modal Systems
The main modal systems are composed from a small set of axioms:
Axiom T: ( P → P)
Axiom D: ( P → ♦P)
Axiom B: (P → ♦P)
Axiom 4: ( P → P)
Axiom 5: (♦P → ♦P)

Main Modal Systems
The main modal systems are composed from a small set of axioms:
Axiom T: ( P → P)
Axiom D: ( P → ♦P)
Axiom B: (P → ♦P)
Axiom 4: ( P → P)
Axiom 5: (♦P → ♦P)
From these axioms, it possible to build the following systems: KT,
KD, KB, K4, K5, K45, KB4, KD4, KTB, KT4, KT5, etc.

Relations between Axiomatic Modal Systems
K
KD K4 K5 KB
KTKDB KD4KD5 K45
KTB KT4 KD45 KB4
KT5

Inclusion of K Axiomatic Systems
For instance, we can demonstrate that KD is included in KT,
We have to deduce ( P → ♦P) from ( P → P)
( P → P)
(¬P → ¬ P)
(¬P → ♦¬P)
T contraposition
(P → ♦P)
( P → ♦P)
Transitivity

Progress
1 Introduction
2 Modal Logics
3 Model Theory
4 Axiomatic Theory

Axioms and Class of Models
Theorem: None of the Axioms D, T, B, 4 and 5 is Valid in the
class of all standards models
Proof: it is enough for each axiom to exhibit a counter-model that
falsiﬁes it. For the axiom D: ( P → ♦P):

1 Let M = (W, R, P) be a standard model in which W = {α},
R = ∅, Pn = ∅ for n 0

R = ∅, Pn = ∅ for n 0
2 M contains one world to which no world is related and at which
every atomic formula is false.

R = ∅, Pn = ∅ for n 0
3 W is a set, R is a binary relation(the empty relation) and P is a
mapping from natural numbers to (empty) subsets of W

R = ∅, Pn = ∅ for n 0
3 W is a set, R is a binary relation(the empty relation) and P is a
mapping from natural numbers to (empty) subsets of W
4 Every β in M such that αRβ is such that |=M
β P and there is
not some β in M such that αRβ and |=M
β P

The necessity Axiom ( P → P) in the following Standard Model:
w1
¬p1
w2
p1
w3
p1
w4
p1R
R
R
R
R
R
RR

Under What Conditions, The necessity Axiom ( P → P) in
the KT system will be valid:
Hypothesis |=M
α P, we have to demonstrate |=M
α P

Hypothesis |=M
α P
1 By deﬁnition |=M
α P iﬀ for every β in M such that αRβ, |=M
β P

Hypothesis |=M
α P
β P
2 Consequently, |=M
α P iﬀ R is reﬂexive, that is to say αRα.

Hypothesis |=M
α P
β P
2 Consequently, |=M
α P iff R is reflexive, that is to say αRα.
In the KT system, the necessity Axiom is valid under the following
condition:
|=C ( P → P), C is a class of models in which R is
Reflexive

Deﬁnition: Let M = (W, R, P) be a standard model, the
relation R, with α, β, γ ∈ W is:

Serial iﬀ for every α in M there is a β in M such that α R β

Reﬂexive iﬀ for every α in M, α R α

Symmetric iﬀ for every α and β in M, if α R β then β R α

Transitive iﬀ for every α, β and γ in M, if α R β and β R γ
then α R γ

Transitive iﬀ for every α, β and γ in M, if α R β and β R γ
then α R γ
Euclidian iﬀ for every α, β and γ in M, if α R β and α R γ
then β R γ

Theorem: the following axioms are valid respectively in the
indicated class of standard models:
D: Serial
T: Reﬂexive
B: Symmetric
4: Transitive
5: Euclidian

Systems Serial Reﬂexive Symmetric Transitive Euclidian
KD x
KT x
KB x
K4 x
K5 x
KDB x x
KD4 x x
KD5 x x
KD45 x x
KB4 x x
KB4 x x
KTB x x
KT4 x x
KT5 x x
KT5 x x x
KT5 x x x
KT5 x x x

Progress
1 Introduction
2 Modal Logics
3 Model Theory
4 Axiomatic Theory

Knowledge and Belief Logic
Two modal operators, B for Belief and K for Knowledge,
Knowledge Properties for an agent λ:
Axioms
1 Axiom 1 (Distribution Axiom, K): (KλP ∧ Kλ(P → Q)) → KλQ
- Same as (Kλ(P → Q)) → (KλP → KλQ))
2 Axiom 2 (Knowledge Axiom or T): (KλP → P)
3 Axiom 3 (Positive Introspection Axiom, 4): (KλP → KλKλP)
4 Axiom 4 (Negative Introspection Axiom): (¬KλP → Kλ¬KλP)
- (¬ P → ¬ P) (♦¬P → ♦¬P)
- (♦¬P → ♦¬P) (♦Q → ♦Q) (axiom 5)

Two modal operators, B for Belief and K for Knowledge,
Knowledge Properties for an agent λ:
Inference Rules
Necessitation: P KλP
Omniscience Logic: if P Q and KλP then KλQ (axiom 1 +
necessitation rule)
The axiomatic theory correspond with the KT45 modal system in
which the accessibility relation is symmetric, reﬂexive and
transitive.

Three-Wise-Men Problem
A king wishes to know whether his three advisors are as wise as
they claim to be.

they claim to be.
Three chairs are lined up, all facing the same direction, with one
behind the other. The wise men are instructed to sit down.

they claim to be.
The wise man in the back (wise man 3) can see the backs of the
other two men.

they claim to be.
other two men.
The man in the middle (wise man 2) can only see the one wise
man in front of him (wise man 1).

they claim to be.
other two men.
The man in the middle (wise man 2) can only see the one wise
man in front of him (wise man 1).
The wise man in front (wise man 1) can see neither wise man 3
nor wise man 2.

they claim to be.

they claim to be.
The king informs the wise men that he has three cards, all of
which are either black or white, at least one of which is
white. He places one card, face up, behind each of the three
wise men.

they claim to be.
The king informs the wise men that he has three cards, all of
which are either black or white, at least one of which is
white. He places one card, face up, behind each of the three
wise men.
Each wise man must determine the color of his own card and
announce what it is as soon as he knows. The ﬁrst to correctly
announce the color of his own card will be aptly rewarded. All
know that this will happen. The room is silent; then, after
several minutes, wise man 1 says: My card is white!.

they claim to be.
We assume in this puzzle that the wise men do not lie, that they
all have the same reasoning capabilities, and that they can all think
at the same speed. We then can postulate that the following
reasoning took place.

Each wise man knows there is at least one white card. If the
cards of wise man 2 and wise man 1 were black, then wise man
3 would have been able to announce immediately that his card
was white.

was white.
They all realize this (they are all truly wise). Since wise man 3
kept silent, either wise man 2’s card is white, or wise man 1’s is.

was white.
At this point wise man 2 would be able to determine, if wise man
1’s were black, that his card was white. They all realize this.

was white.
At this point wise man 2 would be able to determine, if wise man
1’s were black, that his card was white. They all realize this.
Since wise man 2 also remains silent, wise man 1 knows his card
must be white.

If we reduce the problem with two advisors A and B and have the
following axioms:

following axioms:
1 A and B know that each one can see the card of the other one

following axioms:
If A does not have a white card, B knows that A does not have a
white card
- (¬Blanc(A) → KB¬Blanc(A))

following axioms:
white card
A knows that if A does not have a white card, B knows that A
does not have a white card

following axioms:
white card
A knows that if A does not have a white card, B knows that A
does not have a white card
- KA(¬Blanc(A) → KB¬Blanc(A))

following axioms:
2 A and B know that at least one of them has a white card and
each one knows that the other knows it.

following axioms:
A knows that B knows that if A does not have a white card then
B has a white card

following axioms:
B has a white card
- KAKB(¬Blanc(A) → Blanc(B))

following axioms:
B has a white card
B declare that he does not say if he has a white card, then A
knows that B does not knows the color of his card.

following axioms:
B has a white card
B declare that he does not say if he has a white card, then A
knows that B does not knows the color of his card.
- KA¬KBBlanc(B)

If we can reduce the problem with two advisors and have the
following axioms:

following axioms:
KA(¬Blanc(A) → KB¬Blanc(A)) [1]

following axioms:
KAKB(¬Blanc(A) → Blanc(B)) [2]
KA¬KBBlanc(B) [3]

following axioms:
KA¬KBBlanc(B) [3]
[1] (KA(¬Blanc(A) → KB¬Blanc(A)) (KλP → P) [Axiom2]
[4] (¬Blanc(A) → KB¬Blanc(A))

following axioms:
KA¬KBBlanc(B) [3]
[1] (KA(¬Blanc(A) → KB¬Blanc(A)) (KλP → P) [Axiom2]
[2] KAKB(¬Blanc(A) → Blanc(B)) (KλP → P) [Axiom2]
[5] (KB(¬Blanc(A) → Blanc(B))
(Kλ(P → Q) → (KλP → KλQ)) [Axiom1]
[6] (KB¬Blanc(A) → KBBlanc(B))

Omniscience
Contraposition
Transitivity
(KB¬Blanc(A) → KBBlanc(B)) [6]
[7] (¬Blanc(A) → KBBlanc(B))
[8] (¬KBBlanc(B) → Blanc(A))
[1] [2] [4] [5]
[9] (KA(¬KBBlanc(B) → Blanc(A)))
(Kλ(P → Q) → (KλP → KλQ)) [Axiom1]
(KA¬KBBlanc(A) → KABlanc(A))
KA¬KBBlanc(B) [3]
[11] KABlanc(A)

Two modal operators, B for Belief and K for Knowledge, Belief
Properties for an agent λ:
Axioms

Axioms
An agent cannot belief a contradiction: ¬Bλ(False)

Axioms
Positive Introspection Axiom (axiom 4): (BλP → BλBλP)
an agent believes what he believes

Axioms
Positive Introspection Axiom (axiom 4): (BλP → BλBλP)
an agent believes what he believes
(BλBλP → BλP) (converse of the previous axiom)
(BλBαP → BλP)
an agent can belief what another agent believes

Modal Logic Theorem Prover
Some Modal Logic Theorem Provers
MOLTAP, a Modal Logic Tableau Prover :
http://twan.home.fmf.nl/moltap/index.html
AiML.NET Advances in Modal Logic :
http://www.cs.man.ac.uk/ schmidt/tools/
MOLLE : http://molle.sourceforge.net/
LoTREC: possible worlds ﬁnally made accessible :
http://www.irit.fr/Lotrec/
A Theorem Prover for Intuitionistic Modal Logic S5 (IS5):
http://pl.postech.ac.kr/IS5/
MleanTAP: A Modal Theorem Prover (in Prolog) :
http://www.leancop.de/mleantap/

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